Dirichlet μ-Parametric Differential Problem with Multivalued Reaction Term
Abstract
:1. Introduction
- : there exist positive constants with (see (6) for more information about the constant ), such that
- (i)
- in , as ;
- (ii)
- in , as , with and a.e. in Ω;
- (iii)
- , as .
- A priori estimate and embedding results in the Sobolev space (solution space).
- Construction of the sequence (see Definition 1) in a Galerkin context (approximation procedure).
- Analysis of the energy functional associated with (1) (well-posedness and regularity properties).
2. Preliminaries
- (i)
- ;
- (ii)
- if , then ;
- (iii)
- if , then .
- (P1)
- The function is finite, positively homogeneous, subadditive, and satisfieswhere k is the locally Lipschitz constant of ψ.
- (P2)
- .
- (P3)
- The sum rules are as follows:
- (P4)
- For every , one has , and hence,
- (P5)
- If ψ has a local minimum or maximum at , then .
- (P6)
- The Clarke subdifferential is a nonempty, convex, -compact subset of .
- (P7)
- If and are two sequences in X and , respectively, such that and in X and , then we have .
- (P8)
- Mean-value theorem: If ψ is locally Lipschitz on an open neighborhood containing the segment , then there exist and , satisfying
- (i)
- for all ;
- (ii)
- for all ;
- (iii)
- .
3. Energy Functional Analysis
4. Existence Theorems
5. Special Case
6. Conclusions
- : there exist and positive constants with (recall (6)), such that
Author Contributions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
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Ghasemi, M.; Vetro, C.; Zhang, Z. Dirichlet μ-Parametric Differential Problem with Multivalued Reaction Term. Mathematics 2025, 13, 1295. https://doi.org/10.3390/math13081295
Ghasemi M, Vetro C, Zhang Z. Dirichlet μ-Parametric Differential Problem with Multivalued Reaction Term. Mathematics. 2025; 13(8):1295. https://doi.org/10.3390/math13081295
Chicago/Turabian StyleGhasemi, Mina, Calogero Vetro, and Zhenfeng Zhang. 2025. "Dirichlet μ-Parametric Differential Problem with Multivalued Reaction Term" Mathematics 13, no. 8: 1295. https://doi.org/10.3390/math13081295
APA StyleGhasemi, M., Vetro, C., & Zhang, Z. (2025). Dirichlet μ-Parametric Differential Problem with Multivalued Reaction Term. Mathematics, 13(8), 1295. https://doi.org/10.3390/math13081295